MPS-based quantum impurity solvers for DMFTDMFT + DMRG

F. Alexander Wolf and U. Schollwock

Arnold Sommerfeld Center for Theoretical Physics, LMU Munich

Uni Hamburg, 16 Oct 2014

1 / 16

MotivationWhy use DMRG as impurity solver for DMFT?

Where is DMRG better than Quantum Monte Carlo?

◦ EQ: direct access to frequency-dependent observables / access T=0

◦ NEQ: no phase problem . longer simulation times

Where is DMRG better than NRG?

◦ EQ: homogeneous energy resolution / better scaling with number of bands

◦ NEQ: no application of NRG yet

Why hasn’t it been used up to now?

◦ Lanczos: instable and not precise Garcıa, Hallberg & Rozenberg, PRL (2004)

◦ DDMRG: computationally extremely expensive Nishimoto & Jeckelmann, JPhysCondMat, 2

papers (2004), Karski, Raas & Uhrig, PRB (2005), Karski, Raas & Uhrig, PRB (2008)

◦ Chebyshev and Time evolution: much faster and precise Ganahl, Thunstrom, Verstraete,

Held & Evertz, PRB (2014b), Ganahl, Aichhorn, Thunstrom, Held, Evertz & Verstraete, arxiv (2014a), Wolf,

McCulloch, Parcollet & Schollwock, PRB (2014a)

2 / 16

MotivationWhy use DMRG as impurity solver for DMFT?

Where is DMRG better than Quantum Monte Carlo?

◦ EQ: direct access to frequency-dependent observables / access T=0

◦ NEQ: no phase problem . longer simulation times

Where is DMRG better than NRG?

◦ EQ: homogeneous energy resolution / better scaling with number of bands

◦ NEQ: no application of NRG yet

Why hasn’t it been used up to now?

◦ Lanczos: instable and not precise Garcıa, Hallberg & Rozenberg, PRL (2004)

◦ DDMRG: computationally extremely expensive Nishimoto & Jeckelmann, JPhysCondMat, 2

papers (2004), Karski, Raas & Uhrig, PRB (2005), Karski, Raas & Uhrig, PRB (2008)

◦ Chebyshev and Time evolution: much faster and precise Ganahl, Thunstrom, Verstraete,

Held & Evertz, PRB (2014b), Ganahl, Aichhorn, Thunstrom, Held, Evertz & Verstraete, arxiv (2014a), Wolf,

McCulloch, Parcollet & Schollwock, PRB (2014a)

2 / 16

MotivationWhy use DMRG as impurity solver for DMFT?

Where is DMRG better than Quantum Monte Carlo?

◦ EQ: direct access to frequency-dependent observables / access T=0

◦ NEQ: no phase problem . longer simulation times

Where is DMRG better than NRG?

◦ EQ: homogeneous energy resolution / better scaling with number of bands

◦ NEQ: no application of NRG yet

Why hasn’t it been used up to now?

◦ Lanczos: instable and not precise Garcıa, Hallberg & Rozenberg, PRL (2004)

◦ DDMRG: computationally extremely expensive Nishimoto & Jeckelmann, JPhysCondMat, 2

papers (2004), Karski, Raas & Uhrig, PRB (2005), Karski, Raas & Uhrig, PRB (2008)

◦ Chebyshev and Time evolution: much faster and precise Ganahl, Thunstrom, Verstraete,

Held & Evertz, PRB (2014b), Ganahl, Aichhorn, Thunstrom, Held, Evertz & Verstraete, arxiv (2014a), Wolf,

McCulloch, Parcollet & Schollwock, PRB (2014a)

2 / 16

Outline

◦ Matrix product states: efficiently represent many-body wave functions offinite-size systems

◦ From finite-size systems to the thermodynamic limit

◦ Solving equilibrium DMFT using “CheMPS”

◦ Solving nonequilibrium DMFT using a time evolution algorithm

3 / 16

Matrix product statesReview: Schollwock, Annals of Physics (2011)

Product state of local states (compare e.g. Gutzwiller mean-field)

|ψ〉 =

⊗∏i=1...L

(a↑i | ↑i〉+ a↓i | ↓i〉

), aσi ∈ C

=∑σ

( ∏i=1...L

aσi)|σ〉, σ = (σi)

Li=1

. of all possible many body states (superpositions∑

σ cσ|σ〉, cσ ∈ C), those with zeroentanglement are realized

Extend the ansatz by replacing aσi ∈ C with Aσi ∈ Cmi×mi+1 , m1 = 1, mL+1 = 1.

|ψ〉 =∑σ

( ∏i=1...L

Aσi)|σ〉, σ = (σi)

Li=1

. No longer factorizes into product of local states . entangled!

4 / 16

Matrix product statesReview: Schollwock, Annals of Physics (2011)

Product state of local states (compare e.g. Gutzwiller mean-field)

|ψ〉 =

⊗∏i=1...L

(a↑i | ↑i〉+ a↓i | ↓i〉

), aσi ∈ C

=∑σ

( ∏i=1...L

aσi)|σ〉, σ = (σi)

Li=1

. of all possible many body states (superpositions∑

σ cσ|σ〉, cσ ∈ C), those with zeroentanglement are realized

Extend the ansatz by replacing aσi ∈ C with Aσi ∈ Cmi×mi+1 , m1 = 1, mL+1 = 1.

|ψ〉 =∑σ

( ∏i=1...L

Aσi)|σ〉, σ = (σi)

Li=1

. No longer factorizes into product of local states . entangled!

4 / 16

Matrix product statesReview: Schollwock, Annals of Physics (2011)

Product state of local states (compare e.g. Gutzwiller mean-field)

|ψ〉 =

⊗∏i=1...L

(a↑i | ↑i〉+ a↓i | ↓i〉

), aσi ∈ C

=∑σ

( ∏i=1...L

aσi)|σ〉, σ = (σi)

Li=1

. of all possible many body states (superpositions∑

σ cσ|σ〉, cσ ∈ C), those with zeroentanglement are realized

Extend the ansatz by replacing aσi ∈ C with Aσi ∈ Cmi×mi+1 , m1 = 1, mL+1 = 1.

|ψ〉 =∑σ

( ∏i=1...L

Aσi)|σ〉, σ = (σi)

Li=1

. No longer factorizes into product of local states . entangled!

4 / 16

Matrix product statesReview: Schollwock, Annals of Physics (2011)

Product state of local states (compare e.g. Gutzwiller mean-field)

|ψ〉 =

⊗∏i=1...L

(a↑i | ↑i〉+ a↓i | ↓i〉

), aσi ∈ C

=∑σ

( ∏i=1...L

aσi)|σ〉, σ = (σi)

Li=1

. of all possible many body states (superpositions∑

σ cσ|σ〉, cσ ∈ C), those with zeroentanglement are realized

Extend the ansatz by replacing aσi ∈ C with Aσi ∈ Cmi×mi+1 , m1 = 1, mL+1 = 1.

|ψ〉 =∑σ

( ∏i=1...L

Aσi)|σ〉, σ = (σi)

Li=1

. No longer factorizes into product of local states . entangled!

4 / 16

Matrix product statesReview: Schollwock, Annals of Physics (2011)

◦ Manage (truncate) matrix dimensions:

Weight of a Fock state |σ〉 in |ψ〉 (almost) invariant under (truncated) SVD

cσ =∏σi∈σ

Aσi =∏σi∈σ

UσiSσi(V σi

)†

◦ DMRG: Vartiational ground state search (minimize Rayleigh quotient)

∂Aσi∗µν

〈ψ|H|ψ〉〈ψ|ψ〉 = 0

solved efficiently as ansatz is linear in Aσi∗µν .

Important: Short-range interactions ⇒ low entanglement

◦ Time evolutionRepresent exp(−iHt) in Krylov subspace {|t0〉, H|t0〉, H2|t0〉, . . . }.

5 / 16

Matrix product statesReview: Schollwock, Annals of Physics (2011)

◦ Manage (truncate) matrix dimensions:

Weight of a Fock state |σ〉 in |ψ〉 (almost) invariant under (truncated) SVD

cσ =∏σi∈σ

Aσi =∏σi∈σ

UσiSσi(V σi

)†

◦ DMRG: Vartiational ground state search (minimize Rayleigh quotient)

∂Aσi∗µν

〈ψ|H|ψ〉〈ψ|ψ〉 = 0

solved efficiently as ansatz is linear in Aσi∗µν .

Important: Short-range interactions ⇒ low entanglement

◦ Time evolutionRepresent exp(−iHt) in Krylov subspace {|t0〉, H|t0〉, H2|t0〉, . . . }.

5 / 16

Matrix product statesReview: Schollwock, Annals of Physics (2011)

◦ Manage (truncate) matrix dimensions:

Weight of a Fock state |σ〉 in |ψ〉 (almost) invariant under (truncated) SVD

cσ =∏σi∈σ

Aσi =∏σi∈σ

UσiSσi(V σi

)†

◦ DMRG: Vartiational ground state search (minimize Rayleigh quotient)

∂Aσi∗µν

〈ψ|H|ψ〉〈ψ|ψ〉 = 0

solved efficiently as ansatz is linear in Aσi∗µν .

Important: Short-range interactions ⇒ low entanglement

◦ Time evolutionRepresent exp(−iHt) in Krylov subspace {|t0〉, H|t0〉, H2|t0〉, . . . }.

5 / 16

Outline

◦ Matrix product states: efficiently represent many-body wave functions offinite-size systems

◦ From finite-size systems to the thermodynamic limit

◦ Solving equilibrium DMFT using “CheMPS”

◦ Solving nonequilibrium DMFT using a time evolution algorithm

6 / 16

Thermodynamic limit from finite-size system

Extract continuous spectral function ρ(ω) of thermodynamic limit from a finite systemwith discrete energy levels?

Spectral function at T = 0

ρ(ω) =− 1

πImG(ω),

=∑n

wnδ(ω − (En − E0)), wn = |〈En|a†|E0〉|2

of single-particle Green’s function

G(ω) = 〈E0|a1

ω + i0+ − (H − E0)a†|E0〉.

7 / 16

Thermodynamic limit from finite-size systemReview: Lin, Saad & Yang, ArXiv (2013)

Method 1 discrete representation of ρ(ω) =∑n wnδ(ω − En)

ρdiscr(ω) =∑n

wn∆n

χ(ω − En

∆n

), ∆n =

1

2(En+1 − En−1), χ indicator function

8 / 16

Thermodynamic limit from finite-size systemReview: Lin, Saad & Yang, ArXiv (2013)

Method 1 discrete representation of ρ(ω) =∑n wnδ(ω − En)

ρdiscr(ω) =∑n

wn∆n

χ(ω − En

∆n

), ∆n =

1

2(En+1 − En−1), χ indicator function

3 2 1 0 1 2 3/t

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

()

t

V/t =1.0LL=20L=10

Example free SIAMhybrid. t0 = V , hopping ti>0 = twn = |〈En|a†0|E0〉|2 ⇒ ρ(ω) = LDOS

H = −L−2∑i=0

ti(a†iai+1 + h.c.)

Features

◦ for ω = En, rapid pointwiseconvergence to thermodynamiclimit

◦ but: necessitates preciseknowledge of poles and weights

8 / 16

Thermodynamic limit from finite-size systemReview: Lin, Saad & Yang, ArXiv (2013)

Method 1 discrete representation of ρ(ω) =∑n wnδ(ω − En)

ρdiscr(ω) =∑n

wn∆n

χ(ω − En

∆n

), ∆n =

1

2(En+1 − En−1), χ indicator function

3 2 1 0 1 2 3/t

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

()

t

V/t =0.8LL=20L=10

Example free SIAMhybrid. t0 = V , hopping ti>0 = twn = |〈En|a†0|E0〉|2 ⇒ ρ(ω) = LDOS

H = −L−2∑i=0

ti(a†iai+1 + h.c.)

Features

◦ for ω = En, rapid pointwiseconvergence to thermodynamiclimit

◦ but: necessitates preciseknowledge of poles and weights

8 / 16

Thermodynamic limit from finite-size systemReview: Lin, Saad & Yang, ArXiv (2013)

Method 1 discrete representation of ρ(ω) =∑n wnδ(ω − En)

ρdiscr(ω) =∑n

wn∆n

χ(ω − En

∆n

), ∆n =

1

2(En+1 − En−1), χ indicator function

3 2 1 0 1 2 3/t

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

()

t

V/t =1.2LL=20L=10

Example free SIAMhybrid. t0 = V , hopping ti>0 = twn = |〈En|a†0|E0〉|2 ⇒ ρ(ω) = LDOS

H = −L−2∑i=0

ti(a†iai+1 + h.c.)

Features

◦ for ω = En, rapid pointwiseconvergence to thermodynamiclimit

◦ but: necessitates preciseknowledge of poles and weights

8 / 16

Thermodynamic limit from finite-size systemReview: Lin, Saad & Yang, ArXiv (2013)

Method 2 Broadened version of ρ(ω) =∑n wnδ(ω − En)

ρη(ω) =∑n

wnhη(ω − En)

with either hgη(x) = 1√2πη

e− x2

2η2 (Gaussian) or hlη = 1π

1x2+η2

(Lorentzian).

Gaussian

9 / 16

Thermodynamic limit from finite-size systemReview: Lin, Saad & Yang, ArXiv (2013)

Method 2 Broadened version of ρ(ω) =∑n wnδ(ω − En)

ρη(ω) =∑n

wnhη(ω − En)

with either hgη(x) = 1√2πη

e− x2

2η2 (Gaussian) or hlη = 1π

1x2+η2

(Lorentzian).

Gaussian

3 2 1 0 1 2 3/t

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

()

t

V/t =1.0L =40

L=0.2=0.1

Features

◦ uniform convergence for η → 0 andL→∞ requires larger systemsthan pointwise approach

◦ can be generated by expansions insmooth functions, without theprecise knowledge of spectrum andweights

9 / 16

Thermodynamic limit from finite-size systemReview: Lin, Saad & Yang, ArXiv (2013)

Method 2 Broadened version of ρ(ω) =∑n wnδ(ω − En)

ρη(ω) =∑n

wnhη(ω − En)

with either hgη(x) = 1√2πη

e− x2

2η2 (Gaussian) or hlη = 1π

1x2+η2

(Lorentzian).

Gaussian

3 2 1 0 1 2 3/t

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

()

t

V/t =0.8L =40

L=0.2=0.1

Features

◦ uniform convergence for η → 0 andL→∞ requires larger systemsthan pointwise approach

◦ can be generated by expansions insmooth functions, without theprecise knowledge of spectrum andweights

9 / 16

Thermodynamic limit from finite-size systemReview: Lin, Saad & Yang, ArXiv (2013)

Method 2 Broadened version of ρ(ω) =∑n wnδ(ω − En)

ρη(ω) =∑n

wnhη(ω − En)

with either hgη(x) = 1√2πη

e− x2

2η2 (Gaussian) or hlη = 1π

1x2+η2

(Lorentzian).

Gaussian

3 2 1 0 1 2 3/t

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

()

t

V/t =1.2L =40

L=0.2=0.1=0.06

Features

◦ uniform convergence for η → 0 andL→∞ requires larger systemsthan pointwise approach

◦ can be generated by expansions insmooth functions, without theprecise knowledge of spectrum andweights

9 / 16

Outline

◦ Matrix product states: efficiently represent many-body wave functions offinite-size systems

◦ From finite-size systems to the thermodynamic limit

◦ Solving equilibrium DMFT using “CheMPS”

◦ Solving nonequilibrium DMFT using a time evolution algorithm

10 / 16

Chebyshev expansion of spectral functionWeiße, Wellein, Alvermann & Fehske, RMP (2006)/Holzner, Weichselbaum, McCulloch, Schollwock & von Delft, PRB (2011)

Explicit Tn(x) = cos (n arccos(x))

Recursive Tn+1(x) = 2xTn(x)− Tn−1(x)

T0(x) = 1 T1(x) = x

Complete∫ 1

−1

dx√1− x2

Tm(x)Tn(x) ∝ δmn

Expand δ(x−H) in Chebyshev polynomials δN (ω) =∑Nn=1

Tn(ω)√1−ω2

Tn(H)

ρN (ω) = 〈t0|δN (ω −H)|t0〉, |t0〉 = a†|E0〉

Evaluate Tn(H)|t0〉 recursively / “Probe” spectrum of H in vicinity of |E0〉

|tn〉 = 2H|tn−1〉 − |tn−2〉 iterative MPS compression

|t1〉 = H|t0〉 |E0〉 by standard DMRG calculation

11 / 16

Chebyshev expansion of spectral functionWeiße, Wellein, Alvermann & Fehske, RMP (2006)/Holzner, Weichselbaum, McCulloch, Schollwock & von Delft, PRB (2011)

Explicit Tn(x) = cos (n arccos(x))

Recursive Tn+1(x) = 2xTn(x)− Tn−1(x)

T0(x) = 1 T1(x) = x

Complete∫ 1

−1

dx√1− x2

Tm(x)Tn(x) ∝ δmn

Expand δ(x−H) in Chebyshev polynomials δN (ω) =∑Nn=1

Tn(ω)√1−ω2

Tn(H)

ρN (ω) = 〈t0|δN (ω −H)|t0〉, |t0〉 = a†|E0〉

Evaluate Tn(H)|t0〉 recursively / “Probe” spectrum of H in vicinity of |E0〉

|tn〉 = 2H|tn−1〉 − |tn−2〉 iterative MPS compression

|t1〉 = H|t0〉 |E0〉 by standard DMRG calculation

11 / 16

Two-site cluster DCAWolf, McCulloch, Parcollet & Schollwock, PRB (2014a) / CTQMC by Ferrero, Cornaglia, De Leo, Parcollet, Kotliar &

Georges, PRB (2009)

Model: Hole-doped Hubbard model on 2 dimensional square lattice

0.0

0.2

0.4

0.6

0.8

1.0

1.2

A+(

)=

-1Im

G+(

)

(a)

2 1 0 1 2 3 4/D

0.0

0.2

0.4

0.6

0.8

1.0

A-(

)=

-1Im

G-(

)

(b)

this workFerrero (2009)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.41.0

0.5

0.0

0.5

1.0

1.5

2.0

G+(i

n)

(c)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4i n /D

0.8

0.6

0.4

0.2

0.0

G-(

in)

(d)

ReG(i n )ImG(i n )

During Chebyshev recursion, as well as during time evolution, entanglement isgenerated and limits the accessible “time” (number of Chebyshev vectors).

12 / 16

What is the fundmental problem?Wolf, McCulloch, Parcollet & Schollwock, PRB (2014a) / Wolf, McCulloch & Schollwock, ArXiv (2014b)

Must represent hybridization function Λ(t, t′) of impurity problem with veritablequantum degrees of freedom / cannot be analytically evaluated as in CTQMC!

. Choose the least-entangled representation for these quantum degrees of freedom

Hstar = Himp +Hbath +Hhyb,

Hbath =

Lb∑l=1

∑σ

εlc†lσclσ,

Hhyb =

Lb∑l=1

∑σ

(Vlc†0σclσ + H.c.

),

Hchain = Himp +Hpot +Hkin,

Hpot =

Lb∑l=1

∑σ

εlc†lσclσ,

Hkin =

Lb−1∑l=0

∑σ

(Vlc†l+1,σclσ + H.c.

).

Λstar(ω) =

Lb∑l=1

|Vl|2

ω − εl

Λchain(ω) =|V0|2

ω − ε1 −|V1|2

ω − ε2 −· · ·

ω − εLb−1 −|VLb−1|2ω−εLb

,

13 / 16

What is the fundmental problem?Wolf, McCulloch, Parcollet & Schollwock, PRB (2014a) / Wolf, McCulloch & Schollwock, ArXiv (2014b)

Must represent hybridization function Λ(t, t′) of impurity problem with veritablequantum degrees of freedom / cannot be analytically evaluated as in CTQMC!

. Choose the least-entangled representation for these quantum degrees of freedom

Hstar = Himp +Hbath +Hhyb,

Hbath =

Lb∑l=1

∑σ

εlc†lσclσ,

Hhyb =

Lb∑l=1

∑σ

(Vlc†0σclσ + H.c.

),

Hchain = Himp +Hpot +Hkin,

Hpot =

Lb∑l=1

∑σ

εlc†lσclσ,

Hkin =

Lb−1∑l=0

∑σ

(Vlc†l+1,σclσ + H.c.

).

Λstar(ω) =

Lb∑l=1

|Vl|2

ω − εl

Λchain(ω) =|V0|2

ω − ε1 −|V1|2

ω − ε2 −· · ·

ω − εLb−1 −|VLb−1|2ω−εLb

,

13 / 16

Different entanglement in star and chain geometryWolf, McCulloch & Schollwock, ArXiv (2014b)

Model: DMFT for single-band Hubbard model on Bethe lattice

. Compute Green function

0 5 10 15t v

0

0.5

ReG

> (t)

(e)-4 0 4

/v0

1-I

mG

()

(h)

(i)

(ii)

(iii)

Vl

Vl

Vl

Vl ~

. Very different matrix dimension growth in different geometries

1 40

bond

02468

101214

t v

100

200

1 40

bond

02468

101214

20

80

1 40

bond

02468

101214

20

80

(a) (b) (c)0 5 10 15

t v10-1100101102103104

cpu t

ime (

min

)

(g)

14 / 16

Non-equilibrium DMFTWolf, McCulloch & Schollwock, ArXiv (2014b)

Model: single-band Hubbard model on Bethe lattice. quench from atomic limit v = 0 to v = v0

Nonequilibrium Hamiltonian representation by Gramsch, Balzer, Eckstein & Kollar, PRB (2013)

Up to now: exact diagonalization . tmax ∼ 3/v0 for U/v0 = 10Using MPS: tmax ∼ 7/v0 for U/v0 = 10

0 1 2 3 4 5 6 7t v0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

d×

10

-2

Lb =14Lb =16Lb =18Lb =20Lb =24

(a)0 1 2 3 4 5 6 7

t v0

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

Eki

n(v

0)

Lb =14Lb =16Lb =18Lb =20Lb =24

0 1234 56 72.542.522.502.482.46

Eto

t(v

0)

(b)

15 / 16

Non-equilibrium DMFTWolf, McCulloch & Schollwock, ArXiv (2014b)

Model: single-band Hubbard model on Bethe lattice. quench from atomic limit v = 0 to v = v0

Nonequilibrium Hamiltonian representation by Gramsch, Balzer, Eckstein & Kollar, PRB (2013)

Up to now: exact diagonalization . tmax ∼ 3/v0 for U/v0 = 4Using MPS: tmax ∼ 5.5/v0 for U/v0 = 4

0 1 2 3 4 5 6t v0

0

0.02

0.04

0.06

0.08

0.1

d

Lb =10Lb =12Lb =14Lb =16Lb =18

(a)0 1 2 3 4 5 6

t v0

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Eki

n(v

0)

Lb =10Lb =12Lb =14Lb =16Lb =18

0 1 2 3 4 5 6 1.21.11.00.90.8

Eto

t(v

0)

(b)

15 / 16

Non-equilibrium DMFTWolf, McCulloch & Schollwock, ArXiv (2014b)

Model: single-band Hubbard model on Bethe lattice. quench from atomic limit v = 0 to v = v0

Nonequilibrium Hamiltonian representation by Gramsch, Balzer, Eckstein & Kollar, PRB (2013)

With known hybridization function (no self-consistency) Balzer, Li, Vendrell & Eckstein, ArXiv (2014)

0 2 4 6 8 1012t v0

0

0.01

0.02

0.03

0.04

0.05

d Lb =16Lb =18Lb =20Lb =22

(a)0 1 2 3 4 5 6 7 8 9

t v0

00.020.040.060.08

0.10.120.14

d Lb =10Lb =12Lb =16Lb =18

(b)

15 / 16

Summary and Outlook

Summary

◦ Use Chebyshev polynomials to compute spectral functions!

◦ DMFT with DMRG + CheMPS much more efficient than previous MPS methods

◦ Entanglement depends strongly on representation of impurity model . stargeometry favorable

◦ Solving NEQDMFT using MPS allows to access larger times

Outlook

◦ further understand entanglement properties of impurity problems

◦ in equilibrium: apply these results to three-band models . conductivities

◦ in nonequilibrium: treat quenches with correlated initial states

Thanks for your attention!

16 / 16

Summary and Outlook

Summary

◦ Use Chebyshev polynomials to compute spectral functions!

◦ DMFT with DMRG + CheMPS much more efficient than previous MPS methods

◦ Entanglement depends strongly on representation of impurity model . stargeometry favorable

◦ Solving NEQDMFT using MPS allows to access larger times

Outlook

◦ further understand entanglement properties of impurity problems

◦ in equilibrium: apply these results to three-band models . conductivities

◦ in nonequilibrium: treat quenches with correlated initial states

Thanks for your attention!

16 / 16

Summary and Outlook

Summary

◦ Use Chebyshev polynomials to compute spectral functions!

◦ DMFT with DMRG + CheMPS much more efficient than previous MPS methods

◦ Entanglement depends strongly on representation of impurity model . stargeometry favorable

◦ Solving NEQDMFT using MPS allows to access larger times

Outlook

◦ further understand entanglement properties of impurity problems

◦ in equilibrium: apply these results to three-band models . conductivities

◦ in nonequilibrium: treat quenches with correlated initial states

Thanks for your attention!

16 / 16

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